Properties of nucleon in nuclear matter: once more

We calculate the mass and residue of the nucleon in nuclear matter in the framework of QCD sum rules using the nucleon's interpolating current with an arbitrary mixing parameter. We evaluate the effects of the nuclear medium on these quantities and compare the obtained results with the existing theoretical predictions. The results are also compared with those obtained in vacuum to find the shifts in the quantities under consideration. Our calculations show that these shifts in the mass and residue are about $32\%$ and $15\%$, respectively.


Introduction
To analyze the experimental results on the relativistic heavy ion collision held at different experiments such as CERN, the European Organization for Nuclear Research, and BNL, Brookhaven National Laboratory, as well as for better understanding the internal structure of the neutron stars, the in-medium properties of hadrons especially the properties of nucleons at nuclear medium are needed. From the experimental side, there has been a good progress on the in-medium properties of hadrons in recent years. The FAIR (Facility for Antiproton and Ion Research), and CBM (the Compressed Baryonic Matter) Collaboration at GSI intend to study the in-medium effects on the parameters of different hadrons. The Panda Collaboration, on the other hand, aims to concentrate on the properties of the charmed hadrons and study the shifts on their mass and width in nuclear medium [1,2].
From theoretical side, there are dozens of works devoted to the study of the nuclear matter and properties of hadrons especially nucleons at dense medium. In [3], the basic properties of the nuclear matter are determined in the frame work of QCD sum rules as one of the most applicable and attractive tools to hadron physics. This method, then, has been applied to some finite-density problems [4][5][6]. In [4], the authors have used the finite density sum rules to investigate the saturation properties of nuclear matter. In series of papers [7][8][9], T. D. Cohen et al, have applied the QCD sum rules to relativistic nuclear physics and studied the effects of nuclear matter on the mass of the nucleons mostly for the Ioffe current. Only in [9], the authors a bit extend the Ioffe current (β = −1) with β being a general parameter in the interpolating current of the nucleons to −1.15 ≤ β ≤ −0.95 using the mass sum rules. For some studies of nucleon mass shift in nuclear medium for the Ioffe current and properties of other hadrons in dense medium see for instance [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Recently, the QCD sum rules has been used to analyze the residue of the nucleon pole as a function of nuclear density [27] using a special current corresponding to an axial-vector diquark coupled to a quark.
In this article, we calculate the mass and residue of the nucleon using the most general form of the interpolating current in the frame work of QCD sum rules and investigate the shifts in the values of these quantities compared to their vacuum values. We also compare our results on the mass and residue of the nucleon with the existing numerical results obtained via Ioffe current in vacuum. Our results on the mass and residue can be used in the analysis of the experimental results discussed above as well as in theoretical calculations such as computation of the electromagnetic properties and multipole moments of the nucleon and the strong coupling constant of the nucleon to other hadrons in nuclear medium.
The article is organized as follows: In section 2, we obtain the QCD sum rules for the mass and residue of the nucleon in the nuclear matter. Section 3 is devoted to the numerical analyses of the sum rules and our comparison of the results with the existing predictions. Section 4 contains our concluding remarks.

QCD sum rules for the mass and residue of nucleon in nuclear matter
To obtain the sum rules for the mass and residue of nucleon in nuclear matter, the starting point is to consider the following two-point correlation function: where p is the four momentum of the nucleon and |ψ 0 is the nuclear matter ground state. The most general form of the nucleon interpolating current is taken as here a, b, c are color indices, C is the charge conjugation operator and A 1 1 = I, A 2 1 = A 1 2 = γ 5 , A 2 2 = β. As previously said, the parameter β is an arbitrary auxiliary parameter, and β = −1 corresponds to the Ioffe current. The quark flavors for the proton (neutron) are q 1 = u and q 2 = d (q 1 = d and q 2 = u). Here we will adopt the isospin symmetry to treat the proton and neutron as nucleon.
From the general philosophy of the method under consideration, we calculate the above mentioned correlation function from two different windows: in terms of hadronic parameters called as the phenomenological or physical side and in terms of QCD degrees of freedom using the operator product expansion (OPE) at nuclear medium named as QCD or theoretical side. Equating these two sides, we gain QCD sum rules for the mass and residue in nuclear matter. To suppress contribution of the higher states and continuum Borel transformation and continuum subtraction are applied to both sides of the obtained sum rules.

Physical side
In physical side, the correlation function is calculated inserting a complete set of nucleon state with the same quantum numbers as the interpolating current. After performing integral over four-x, we get where dots represents the contributions of higher states and continuum and m * N is the modified mass of the nucleon in nuclear matter. The matrix element of the interpolating current between the nucleon ground state and the baryonic state is parametrized as here λ * N is the modified residue or the coupling strength of the nucleon current J(x) to the nucleon quasi-particle in the nuclear matter and u(p, s) is their positive energy Dirac spinor. Using Eq. (4) in Eq. (3), we get Considering the interactions between the nucleon and the nuclear matter, the physical side of the correlation function takes the modified form where Σ µ ν and Σ s are vector and scalar self-energies of the nucleon in nuclear matter, respectively [10]. In general, we can write where Σ ν and Σ ′ ν are constants and u µ is the four velocity of the nuclear medium. Here we neglect Σ ′ ν due to its small contribution (see also [10]). Apart from the vacuum QCD calculations, the four-velocity of the nuclear matter is new concept that causes extra structures to the correlation function. We shall work in the rest frame of the nucleon with u µ = (1, 0). Substitution Eq. (7) into Eq. (6), the physical side of the correlation function becomes which can be written in terms of three different structure as where p 0 is the energy of the quasi-particle, I is the unit matrix and Here Using the Borel transformation with respect to p 2 , we getB where M 2 is the Borel mass parameter, we shall find its working region in Section 3.

QCD side
The QCD side of the correlation function can be calculated using (OPE) in deep Euclidean region. This function can also be written in terms of different structures in QCD side as Each Π QCD i function where i = p, u and I, can be written in terms of dispersion integral as where ] are the spectral densities. Using the explicit form of the interpolating current in correlation function of Eq. (1) and contracting out all quark pairs via Wick's theorem, we find where S ′ = CS T C and S u,d are light quarks propagators. In coordinate-space, the light quark propagator at the nuclear medium has the following form in the fixed-point gauge [14]: where ρ N is the nuclear matter density and the first and second terms are the expansion of the free quark propagator to first order in the quark mass (perturbative part), and the third and forth terms are the contributions due to the background quark and gluon fields (non-perturbative part), respectively. When using Eq. (15) in Eq. (14), we will have the ground-state matrix elements of the quark and gluon operators as [14] To proceed, we need to define the quark and gluon and mixed condensates in nuclear matter. The matrix element q aα (x)q bβ (0) ρ N is projected out as [14,28] The quark-gluon condensate in nuclear matter is written as where t A ab are Gell-Mann matrices and D µ = 1 2 (γ µ D+ Dγ µ ). The matrix element of the four-dimension gluon condensate can also written as where we neglect the last term in this equation because of its small contribution. The various condensates in the above equations are defined as [8,14] qγ µ q ρ N = q uq ρ N u µ , After lengthy calculations, for the Π QCD i functions in QCD side, we get the following expressions in Borel scheme: and Having calculated both the physical and QCD sides of the correlation function, now, we equate these two sides for all structures to find the corresponding QCD sum rules. For instance, in the case of the structure p, we have To find the mass sum rule, we eliminate the λ * 2 N in the above equation, as a result of which we get

Numerical results and discussion
This section is devoted to the numerical analysis of the sum rules for the mass and residue obtained in the previous section at nuclear matter. We discuss how the results in dense Input Parameters Table 1: Numerical values for input parameters [8,9,12,14].
medium deviate from those obtained via vacuum sum rules. For this aim, we need the numerical values of the quark masses as well as the in-medium quark-quark, quark-gluon and gluon-gluon condensates that are calculated in [8,9,12,14]. Each condensate at dense nuclear medium ( O ρ N ) is written in terms of its vacuum values ( O 0 ) and its value between one-nucleon states ( O N ) at the low nuclear density limit, i.e., We collect the numerical values of the input parameters in Table 1.
Looking at the sum rules for the physical quantities under consideration we see that they include three auxiliary parameters, namely, continuum threshold s 0 , Borel mass parameter M 2 and general parameter β that should be fixed at this point. The standard criteria in QCD sum rules demand that the physical quantities show good stability with respect to these quantities at their working regions. As the mass sum rule is the ratio of two sum rules (see Eq. 29) including these auxiliary parameters, it may not lead to a reliable region. For this reason, we use the sum rule for the residue to find the reliable regions for the helping parameters. The working region for the Borel mass parameter is found demanding that not only the contributions of the higher states and continuum are sufficiently suppressed and the ground state constitutes a large part of the whole dispersion integral, but also the perturbative part exceeds the non-perturbative one and the contributions of the operators with higher dimensions are small, i.e., the OPE converges. Our numerical calculations lead to the interval 0.8 GeV 2 M 2 1.2 GeV 2 for the Borel mass squared. The continuum threshold is not totally arbitrary but it depends on the energy of the first excited state with the same quantum numbers as the interpolating current. Our numerical calculations depict that in the interval s 0 = (1.5 − 2.0) GeV 2 the physical results depend weakly on this parameter.  To find the working region for the general parameter β, we look at the variation of the residue of the nucleon with respect to this parameter. To better cover the whole range −∞ β ∞, we plot the the residue with respect to x = cosθ, where β = tanθ at fixed values of the continuum threshold and Borel mass parameter for both nuclear medium and vacuum in figure 1. From this figure, we see that in the intervals −1 x −0.5 and 0.5 x 1 the residues λ * N and λ N are practically independent of this parameter. Moreover, the results of residues depend on continuum threshold very weakly in these intervals. Note that the Ioffe current corresponding to x ≈ −0.71 is included by these intervals. Here, we should mention that, as also we said before, since the mass sum rule  is the ratio of two sum rules, the unstable points of two sum rules in nominator and denominator coincide and cancel each other. Such that, the masses in nuclear matter and vacuum show roughly good stabilities with respect to x in the whole −1 x −1 region (see figure 2).
Having calculated the working regions, now, we discuss the variations of the masses and residues both in nuclear matter and vacuum with respect to the variations of the auxiliary parameters and look for the shifts in these parameters due to the nuclear medium by comparison of the results obtained in the nuclear matter as well as the vacuum. For this aim, in figures 3 and 4, we depict the variations of the residues and masses in the presence of nuclear matter and vacuum with respect to the Borel mass parameter at different fixed values of the β and s 0 picked from their working regions. These figures also indicate that the physical quantities under consideration vary weakly with respect to the helping parameters in their working regions.
Obtained from figures 3 and 4, we depict the average values of the residues squared and masses of the nucleon both for the nuclear medium and vacuum in Table 2. We also compare our results with the existing results obtained via the Ioffe current using the vacuum sum rules in this table. From this table, we conclude that the average values for the residue 0.723 ± 0.122 1.045±0.076 0.0009 ± 0.0004 0.0011 ± 0.0005 Vacuum Sum Rules [29] -0.985 -0.0012 ± 0.0006 Table 2: Average values of the masses and residues squared in nuclear matter and vacuum obtained from sum rules analysis and the comparison of the results with the existing results of the vacuum sum rules for the Ioffe current. squared and mass when ρ N → 0 are consistent with the values obtained using the Ioffe current and vacuum sum rules [29] within the errors. We also see that the average values of those quantities in nuclear matter show considerable shifts compared to the vacuum results. To see better how the results of the residue and mass in nuclear matter deviate from those of the vacuum, we depict the variations of the ratios of λ * N /λ N and m * N /m N as well as the percentages of the shifts with respect to the Borel mass squared in figures 5 and 6 at different fixed values of the general parameter β and the continuum threshold s 0 . With a quick glance at these figures, we observe that the mass and residue of the nucleon show considerable shifts from their vacuum values and the shifts are negative. In the case of the residue, the shift grows roughly increasing the value of the Borel mass parameter. However, in the case of mass the shift deceases considerably when we increase the value of the Borel mass parameter in its working region. Our numerical results show that, in average, the values of the residue and mass decrease about 15% and 32%, respectively compared to their values in vacuum.

Conclusion
In the present work, we studied some properties of nucleon in the nuclear matter using the QCD sum rules in nuclear medium. In particular, we calculated the mass and residue of the nucleon in nuclear medium and looked at the shift of the results compared to their vacuum values. Using the most general form of the interpolating current for the nucleon, we extended the previous works on the mass of the nucleon discussed in the body text which mainly use the Ioffe current. We also extended the recent study [27] on the residue of the nucleon pole with a special current corresponding to an axial-vector diquark coupled to a quark by considering the general interpolating current. We found the working regions for the three main auxiliary parameters entering the sum rules using the obtained QCD sum rule for the residue. Using the obtained working regions for the continuum threshold, Borel mass parameter and the general parameter β entering to the general interpolating current, we depicted the variations of the physical quantities under consideration with respect to the variations of the auxiliary parameters. We observed considerable negative shifts in the values of the mass and residue of the nucleon in nuclear matter compared to their values in vacuum. The results of the residue and mass reduce about 15% and 32%, respectively due to the nuclear medium. Our results can be used in analysis of the results of the heavy ion collision experiments as well as in understanding the internal structures of the heavy-dense objects like neutron stars. The obtained result for the residue in nuclear matter can also be used in theoretical determination of the electromagnetic properties of the nucleon and its strong couplings to other hadrons in the nuclear medium.